The Solid-State Hard-Ball Model and Young’s Modulus

Tungsten is a ferri-type paramagnetic metal in that its paramagnetism obeys the Neél ferrimagnetic formula in fitting susceptibility data χ from low of 5K to high of 300K [1] and 1800K [2]. The magnetic susceptibility data of VI B elements, Cr, Mo, and W are increasing with temperature, quite different from that of V B elements, V, Nb, and Ta that have decreasing trend with temperature [3]. This means that the electronic configurations of elements play important role in their properties, e.g., Young’s moduli of VI B are greater than that of V B. χ inverse is equal to the inverse of the diamagnetic χ Curie [1], plus the inverse of the paramagnetic χ Pauli, and plus the inverse of the diamagnetic χ _{lattice/phonon}. The inverse of the diamagnetic χCurie has the form of T/C where T is in K, and the negative C is also in K; the inverse of the paramagnetic χPauli is independent of T, and hence a constant; while the inverse of the diamagnetic χ_{lattice/phonon} has the form of -b/(T–θ_p) in which b and θp are respectively positive and negative parameters. According to the above discussion, for W χ increases with T, the inverse of χ decreases with T, therefore χ_{Curie-diamagnetic} dominates χ since it is a negative value; while for V B elements χlattice/phonon dominates χ. The diamagnetic χ_Curie comes from the demagnetizing field in the vacancy (0.32) of W in-between the hard ball arrangement (0.68) of atoms in the lattice [1]. The ratio of 0.32 to 0.68 is near the absolute value of that of atoms with the anti-direction (-0.985) to paradirection spins (1.985) by the calculation in the Neél ferrimagnetic formula. Thus one can say that VI B elements obey more likely the hard ball model than do VB elements. It also tells that those with high Young’s modulus fit the hard ball model more than those with low Young’s modulus and the phonon effect is greater with lower modulus than with higher modulus, and reversely in this case.

1. Chen SK, Jou BW, Tsai YT (2018) Ferri-paramagnetism of tungsten and its carbide cermets for elevated temperature application. Chinese Journal of Physics 56: 303-307.
2. Kriessman CJ (1953) The high temperature magnetic susceptibility of V, Nb, Ta, W, and Mo. Rev Mod Phys 25: 122-126.
3. Kittel C (1996) Introduction to solid state physics, (7th edn), John Wiley, Hoboken, NJ, USA, 14: 437.